Speed Control for PMSM Servo System Using Model Reference Adaptive Control and Extended

State Observer

Xiaodi Li and Shihua Li* *School of Automation, Southeast University & Key laboratory of Measurement and Control of Complex Systems of

Engineering, Ministry of Education, Nanjing 210096, P.R.China

Abstract

In this paper, the speed regulation problem for permanent synchronous motor (PMSM) system under vector control framework is

studied. A model reference adaptive controller (MRAC) based on Lyapunov stability theory is first designed. Noticing the standard MRAC method provides a poor disturbance rejection performance in the case of strong disturbances, a composite control method which combines MRAC method and disturbance estimation method, called the MRAC+ESO method, is proposed. An extended state observer (ESO) is introduced to estimate the lumped disturbances. The obtained estimated value acts as a feedforward compensation term to the MRAC controller. The stability analysis of the composite control method is given. Simulation and experimental comparison are presented to show the effectiveness of the proposed control method. Key words: Model reference adaptive control(MRAC), Permanent magnet synchronous motor(PMSM), Speed regulation system, Extended state observer(ESO)

I. INTRODUCTION Permanent magnet synchronous motor (PMSM) has been

widely used in motion control applications due to its excellent features such as simple structure, high efficiency, high power density and friendly maintenance. Many industrial applications of servo systems require high performance such as fast response, low overshoot and insensitivity to disturbance and parameter variation. However, PMSM is nonlinear plant with states coupling. Though there are methods to make the states decoupled approximately such as setting the referenced direct axis current to be zero, it is still difficult for conventional linear control methods, including the proportional-integral (PI), to achieve a high-precision performance [1]. To enhance the control performance, more advanced control methods have been introduced to PMSM servo system, e.g., adaptive control [2]-[6], [19], disturbance estimation based control [7]-[8], sliding model control [9]-[11], finite-time control [17], predictive control [13]-[14], fuzzy control [15], [18]-[19], and neural network control [1], [12], [15]-[16], etc. These methods have improved the performance of the PMSM servo system from different aspects.

Among these methods, the adaptive control method has been widely used in the situations where the system parameters are inaccurately known or the system is operated over a wide range of different operating conditions. In such situations the usual controller with fixed gains may not provide a satisfactory performance. one concept is direct adaptive control which

identifies the parameters of the unknown system and uses the current parameter to synthesize suitable control gains. This approach is easy to understand and implement, but it needs to identify the system parameters online which may not be allowed in some situations. For example, in many application cases, PMSM system may not be allowed to add a sufficiently exciting periodical input signal for online inertia identification [3].

Another concept is the model reference adaptive control (MRAC) approach. It employs a reference model to generate a reference output. The adaptive laws derived by means of the Lyapunov stability theory modify the parameters of the controller without the necessity of a sufficiently exciting system input signal [21]. There have been some cheering research reports on the application of the MRAC method to the motion control system. In [22], MRAC technique is employed for velocity and currents loops of induction motor control systems. In [24], MRAC method is applied to the outer speed loop of the permanent magnet brushless DC motor. In [23], MRAC algorithm for robust control of PMSM is proposed and applied to the three control loops including a speed loop and two current loops. [25] and [27] combine MRAC method and fuzzy control method, MRAC method and variable structure control method in the speed loop to improve the performance of DC motor system and induction motor system, respectively.

In industrial situation, PMSM systems always face with different disturbances, e.g., friction force, load disturbances and unmodeled dynamics [29]. The conventional control methods

may not react directly and fast to reject these disturbances although these control methods can finally suppress them through feedback regulation [3]. Although usually the disturbances can not be measured, there is still one efficient way to fast reject disturbances. That is, estimating disturbances and feedforward compensating them based on the estimated value of them. Disturbance estimation based control is one of the most useful methods and has been well used in different applications, e.g., robotic system [36], spacecraft system [35], PMSM system [28], general motion control system [37], etc. Extended state observer (ESO) technique is one of the most useful observers in the disturbance observer field. Under this framework, many control methods based on feedforward compensation techniques for disturbances have been developed for PMSM systems [28]-[29].

In this paper, the speed regulation problem for PMSM system is studied. The target here is to pursue a high closed loop performance with good adaptation ability for different working conditions as well as strong robustness ability against disturbances and uncertainties. First, a MRAC method is developed for the speed loop of PMSM system. Second, to improve disturbance rejection ability, a composite speed control method using MRAC method and disturbance estimation based on compensation technique is proposed for PMSM servo system. The simulation and experimental comparison results of both methods are also presented.

This paper is organized as follows. Section Ⅱ introduces the PMSM model. The MRAC-based speed control scheme is proposed in Section Ⅲ. Section Ⅳ shows the MRAC+ESO composite control strategy. A conclusion is given in Section Ⅴ.

II. THE MATHEMATICAL MODEL OF PMSM

The model of the surface mounted permanent magnet synchronous motor is expressed as follows [30]:

0

0

s dp

d dp f qs

q p q

t L

R unL Li i

n uRi n iL L L

K TBJJ J

ω

ψω

ω ω

− = − − − +

−−

(1)

where sR the stator resistance, di and qi the d and q

axes stator currents. du and qu the d and q axes stator

voltages, pn the pole pairs number, L the stator

inductance, ω the rotor angular velocity, fψ the flux

linkage, tK the torque constant, J the rotor inertia, B the

viscous friction coefficient, and LT the load torque.

The design procedure is based on the framework of vector control, where a structure of cascade control loop including a

speed tracking loop and two current tracking loops are employed. Here two PI controllers are adopted in the two current loops. As shown in Fig. 1, the rotor angular velocity

can be obtained from the position sensor. The current di and

qi can be calculated from iα and iβ by clarke and park

transformation. Usually, the reference current is set as 0di∗ = .

The reference current qi∗ is determined by the speed loop

controller. In this paper, we concentrate on the design of the speed loop controller.

From (1), the torque equation of PMSM system can be written as

( )

( )

( )

t Lq

t tLq q q

tq

q

K TBiJ J JK KTBi i iJ J J J

KB i d tJ J

a bi d t

ωω

ω

ω

ω

∗ ∗

∗

∗

= − −

= − − − −

= − + +

= − + +

(2)

where BaJ

= , tKbJ

= . ( ) ( )tLq q

KT i id tJ J

∗= − − − can

be considered as the lumped disturbances including the load torque disturbance and the tracking error of q axis current loop. Based on the model shown in (2), a speed loop controller will be designed to improve the adaptation and disturbance rejection ability of PMSM speed system.

III. MRAC CONTROLLER DESIGN FOR PMSM A. Control Design

The MRAC structure is shown in Fig. 2, where ( )mG s is

the reference model, and ω∗ is the reference speed. In Fig. 2, the “Generalized PMSM” represents the two current loops that include the PMSM and other components [3].

The reference model is chosen as:

m m m ma bω ω ω∗= − + (3)

where 0ma > , 0mb > are parameters for this reference

model.

Fig. 1. Block diagram of the PMSM control system

Supposing the control law is described as ( ) ( )qi h t k tω ω∗ ∗= + (4)

where ( )k t the variable feedforward gain and ( )h t the

variable feedback gain. Substituting (4) into (2), yields

( )( ) ( )a bh t bk tω ω ω∗= − − +

(5) Here, define the speed tracking error and parameter error

as follows:

me ω ω= − (6)

( )( )

k k th h t

φ∗

∗

−= −

(7)

where mbkb

∗ = , ma ahb

∗ −= .

Then, differentiating (6) along system (3) and (5) yields ( ) ( )m m m

TTm

e a b a bh t bk t

a e b

ω ω ω ω ω

φ ω ω

∗

∗

= − + + − −

= − +

(8)

Considering the Lyapunov function

12

2

12 2

TbV eγ

φ φγ

= +

(9)

where 1 0γ > , 2 0γ > .

The differentiation of (9) along the trajectory of (8) yields

( )( )

1

2

12

2

T

T Tm

V ee b

k ta e be b

h t

γφ φ

γ

γωφ φ

γω

∗

= +

= − + −

(10)

Choose

( )( )

( )( )

1

2

1

1k k tk t

eh h th t

γ ωω

γ

∗∗

∗

− = + −

(11)

Then it can be obtained that 22 2 0T

m mV a e b a e bφ φ φ= − − = − − ≤ (12)

Therefore, according to the Lyapunov stability theorem, it

can be concluded that the closed loop system is asymptotically stable.

B. Simulation and Experimental Results To demonstrate the efficiency of the MRAC method, some

simulation and experiments on PMSM servo system have been done. Two methods, i.e., MRAC and PI, are both tested on PMSM system.

The tested motor in simulation and experiments is the 750W PMSM designed for servo applications. The PMSM parameters are listed in Table 1: stator resistance 1.74sR = Ω , stator

inductances 0.004d qL L L H= = = , number of pole pairs

4pn = , moment of inertia 4 21.78 10J Kg m−= × ⋅ , viscous

coefficient 57.4 10 /B N m s rad−= × ⋅ ⋅ , rotor flux 0.402f Wbψ = , The PI parameters of both current loops are:

42pK = , 2600iK = . The saturation limit of q -axis

reference current is 9.42A± . Table 1

The parameters of PMSM

Type: EMJ08ADB11

Rated power NP 750W

Stator inductance L 0.004H

Rated voltage NU 200V

Number of Pole pairs pn 4

Rated current NI 4.71A

Stator resistance sR 1.74Ω

Rated speed Nn 3000 / minr

Rotor flux fψ 0.402Wb

PMSM inertia J 4 21.78 10 Kg m−× ⋅

Viscous coefficient B 57.4 10 /N m s rad−× ⋅ ⋅ Simulation Results: The parameters for PI speed controller

are: 0.2pK = , 40iK = , for MRAC speed controller are:

100m ma b= = , 1 2 0.015γ γ= = . Fig. 3(a) and (b) show the

response curves of MRAC-based controller and PI controller in the case of 1000 rpm reference speed. In the simulation, a load torque 2LT N m= ⋅ is applied at 0.6t s= . The speed and

qi∗ response curves are also given in Fig. 3 (c) and (d).

Fig. 2. Block diagram of the standard MRAC method for PMSM system

Experimental Results: To evaluate the performance of the MRAC method, the experimental setup system for the speed control of PMSM has been built. The configuration of it and the experimental test setup are shown in Fig. 4 and Fig. 5. The whole speed control algorithms including the SVPWM are implemented by the program of the DSP TMS320F2808 with a clock frequency of 100MHz. The control algorithm is implemented using C-program. The speed-loop and current-loop sampling periods are 250 sµ and 60 sµ , respectively. The PMSM is driven by a three-phase pulse width modulation (PWM) inverter with an intelligent power module (IPM) with a switching frequency of 10kHz. The phase currents are measured by the Hall-effect devices and are converted through two 12-bit analog to digital converters. An incremental position encoder of 2500 lines is used to measure the rotor speed and absolute rotor position.

The parameters of speed-loop controller are selected as: 100m ma b= = , 1 2 1γ γ= = . Some experimental results are

shown in Figs. 6, 7 and 8 when the reference speed is given as 500 rpm, 1000 rpm and 2000 rpm, respectively. Compared with the PI controller, it seems that the MRAC-based

controller does not have much advantage. Then the rotor inertia J is increased to 11J . Experimental tests have also been done to evaluate the performance of the two control methods working in this situation. The results are shown in Figs. 9, 10 and 11. From these results, it can be observed that the performance of the controller with fixed gains (PI) becomes worse when the inertia of system is changed. However, the MRAC controller with variable gains can adapt to the situation and get a high performance.

Fig. 4. The configuration of experimental system

Fig. 3. Simulation response curves under MRAC and PI controllers: (a) speed. (b) qi

∗ . (c) speed response curves in the presence of load

torque disturbance. (d) qi∗ response curves in the presence of load torque disturbance.

Tests also have been done to evaluate the performance in the presence of sudden load torque disturbance. When PMSM

system is running at a steady speed, a load torque LT is

added suddenly and removed after some duration. The experimental results are given in Figs. 12, 13 and 14. It can be observed that the MRAC method shows a poor disturbance rejection ability which may limit its application. In the following section, a composite method will be proposed to improve the disturbance rejection ability of closed loop system using disturbance estimation theory.

Fig. 5. The experimental test setup

Fig. 6. Experimental response curves when the reference speed is 500 rpm: (a) speed and (b) qi

∗

Fig. 8. Experimental response curves when the reference speed is 2000 rpm: (a) speed and (b) qi

∗

Fig. 7. Experimental response curves when the reference speed is 1000 rpm: (a) speed and (b) qi

∗

IV. MRAC+ESO COMPOSITE CONTROLLER DESIGN FOR PMSM

A. Control Design In the design procedure presented above, the influence of

disturbance ( )d t is ignored. In fact, the load torque

disturbance and the unmodeled dynamics are unavoidable in real industry situations. These disturbances may degrade the performance of the system. For the speed loop of PMSM system, the lumped disturbances here include the load torque disturbances and the tracking error of q axis current loop.

If the lumped disturbances are estimated and feed-forward based on the observed value, the influence caused by the disturbance ( )d t will be restrained. In this paper, an ESO

is added to the MRAC controller. It regards the disturbances

( )d t as a new state of system and estimates both the states

and the disturbances. The detail principle of ESO can be found in [31]-[32].

According to (2), define 1x ω= , ( )2x d t= , then (2) can

be written as

1 2 1

2 ( )x x ax biqx c t

∗ = − +

=

(13)

where ( )c t the derivative of ( )d t .

Fig. 10. Experimental response curves when J is increased to 11J and reference speed is 1000 rpm: (a) speed and (b) qi

∗

Fig. 11. Experimental response curves when J is increased to 11J and reference speed is 2000 rpm: (a) speed and (b) qi

∗

Fig. 9. Experimental response curves when J is increased to 11J and reference speed is 500 rpm: (a) speed and (b)

qi∗

Then, a second-order linear ESO for system (13) is

Fig. 13. Experimental response curves in the presence of sudden load torque disturbance in the case of 1000 rpm: (a) speed (PI). (b) qi

∗ (PI). (c) speed (MRAC). (d) qi∗ (MRAC).

Fig. 12. Experimental response curves in the presence of sudden load torque disturbance in the case of 500 rpm: (a) speed (PI). (b)

qi∗ (PI). (c) speed (MRAC). (d) qi

∗ (MRAC).

Fig. 14. Experimental response curves in the presence of sudden load torque disturbance in the case of 2000 rpm: (a) speed (PI). (b) qi

∗ (PI). (c) speed (MRAC). (d) qi∗ (MRAC).

designed as follows:

( )( )

1 2 1 1 1

22 1 1

2z z ax p z x biq

z p z x

∗ = − − − +

= − −

(14)

where 1z the estimation of speed 1x , and 2z the

estimation of 2x , p− the desired pole of ESO with

0p > . The block diagram of the composite control method based

on MRAC and ESO is shown in Fig. 15. It can be observed that a MRAC controller and an extended state observer are employed to construct the composite MRAC+ESO structure. Under this control method, the composite control form is

( )ˆqu i d t∗= − (15)

B. Stability Analysis Assumption 1. The disturbance ( )d t is bounded, and it

satisfies ( )=0d t .

Lemma 1 [34]. Let V: [0, ) nR R∞ × → be a continuously differentiable function such that

( ) ( ) ( )1 2,x V t x xα α≤ ≤

3( , , ) ( )V V f t x u W xt x

∂ ∂+ ≤ −

∂ ∂, ( ) 0x uρ∀ ≥ >

( , , ) [0, ) n mt x u R R∀ ∈ ∞ × × , where 1α , 2α are class K∞

functions, ρ is a class K function, and ( )3W x is a

continuous positive definite function on nR . Then, the system

( , , ), ,n mx f t x u x R u R= ∈ ∈ is input-to-state stable with 1

1 2γ α α ρ−= .

Lemma 2 [34]. If the following system

( ), ,x f t x u= (16)

satisfies the conditions: 1. system (16) is globally input-state stable; 2. lim 0

tu

→∞= , then the states of system (16) are

asymptotically convergent to zero. Substituting (15) into (2) and combining with (3), the speed

tracking error is given by

( ) ( )( ) ( )

( ) ( )

ˆ

ˆ

m m m

TTm

e a b a bh t bk t

b d t d t

a e b b d t d t

ω ω ω ω ω

φ ω ω

∗

∗

= − + + − −

− −

= − + − −

(17)

For system (17), also choose Lyapunov function as

12

2

12 2

TbV eγ

φ φγ

= +

(18)

where 1 0γ > , 2 0γ > .

The differentiation of (18) along the trajectory of (17) yields

( ) ( )2 2ˆmV a e be d t d t b φ = − − − −

(19)

(1) If mb a≥ , then (19) can be written as

( ) ( )

2 2 2 2

12 2 2 2

(| | ) (| | )2 2

ˆ( ) (| | ) | |

m m

m

a aV e e

b a b e d t d t

φ φ

φ φ

≤ − + − +

− − + + −

(20)

Supposing that

( ) ( )1

2 2 2 2 2 ˆ(| | ) (| | ) | | 02ma e b e d t d tφ φ− + + + − ≤

(21) That is to say

( ) ( ) ( )12 2 2

ˆ2 | || |

m

b d t d te

aφ

−+ ≥

(22)

In this case, it can be obtained that 2 2 2

2 2

(| | ) ( )2

| | ( ) 02 2

mm

m m

aV e b a

a ae b

φ φ

φ

≤ − + − −

= − − − ≤

(23)

(2) If mb a< , then (19) can also be written as

( ) ( )

2 2 2 2

12 2 2 2

(| | ) (| | )2 2

ˆ( ) | | (| | ) | |m

b bV e e

a b e b e d t d t

φ φ

φ

≤ − + − +

− − + + −

(24)

Supposing that

( ) ( )1

2 2 2 2 2 ˆ(| | ) (| | ) | | 02b e b e d t d tφ φ− + + + − ≤

(25) That is to say

( ) ( )1

2 2 2 ˆ(| | ) 2 | |e d t d tφ+ ≥ −

(26)

Then it can be obtained that 2 2 2

2 2

(| | ) ( ) | |2

( ) | | 02 2

m

m

bV e a b e

b ba e

φ

φ

≤ − + − −

= − − − ≤

(27)

From the above analysis for both cases, by Lemma 1 it can be obtained that system (17) is input-state stable.

Consider the situations where the lumped disturbances

( )d t satisfy Assumption 1. According to the analysis in [33],

Fig. 15. Block diagram of MRAC+ESO method for PMSM system

the ESO states ( ) ( )1 1z t x t→ and ( ) ( )2 2z t x t→ . Then

the estimation of the disturbance ( ) ( )2d t z t= satisfies:

( ) ( )ˆlim 0t

d t d t→∞

− = (28)

Treating ( ) ( )ˆd t d t− as the system input and together

with (28) and Lemma 2, it can be concluded that the closed loop system is asymptotically stable.

C. Simulation and Experimental Results To evaluate the performance of the proposed composite

method, simulation and experiments on PMSM system also have been performed.

Simulation Results: The parameters for MRAC+ESO speed controller are: 100m ma b= = , 1 2 0.015γ γ= = , 450p = .

The speed response curves under the MRAC+ESO are shown in Fig. 16. From the curves, it can be observed that the MRAC method and the MRAC+ESO method almost have the same speed response without load torque disturbance. However, with a sudden load torque disturbance 2LT N m= ⋅ applied at

0.6t s= , it can be seen that the system using MRAC+ESO provides a less speed fluctuation against disturbance.

Experimental Results: In the experiments, the MRAC+ESO parameters of speed-loop are selected as: 100m ma b= = ,

1 2 1γ γ= = , 250p = . The speed responses under

MRAC+ESO method are shown in Figs. 17, 18 and 19 when the reference speed is given as 500 rpm, 1000 rpm and 2000 rpm, respectively. From the response curves, the MRAC and

MRAC+ESO method almost have the same speed response which corresponds to the simulation.

Then the inertia J of PMSM system is increased to11J . Experiments are performed to test the adaptability of the MRAC+ESO method. The results are shown in Figs. 20, 21 and 22.

Tests also have been done to evaluate the disturbance rejection ability of MRAC+ESO method in the presence of sudden load torque disturbance. The results are shown in Figs. 23, 24 and 25. Compared with the MRAC method, the proposed MRAC+ESO method has smaller speed decrease and shorter recovery time while maintaining a good dynamic performance at the same time.

From all the experimental results above, it can be observed that though the MRAC+ESO method degrades the adaptation ability, it can obtain a better performance of tracking and disturbance rejection compared with PI and MRAC method.

The performance indexes of each control method at different conditions are shown in Table 2.

V. CONCLUSION In this paper, the design of a speed controller based on

MRAC for the PMSM has been investigated. This method has shown a better adaptation ability compared with PI method. In order to improve the ability of disturbance rejection of the controller, a composite controller which combines MRAC method and an ESO together has been proposed. Simulation and experimental results have validated that the composite method can obtain a satisfying performance with faster transient response and better disturbance rejection ability.

ACKNOWLEDGEMENTS

This work was supported by New Century Excellent Talents in University (NCET-10-0328), Science Foundation for Distinguished Young Scholars of Jiangsu Province under Grant BK20130018 and High-level Talents Program in Six Industries of Jiangsu Province under Grant DZXX-30.

Fig. 16. Simulation response curves under MRAC+ESO and MRAC controllers: (a) speed. (b) qi

∗ . (c) speed response curves in the

presence of load torque disturbance. (d) qi∗ response curves in the presence of load torque disturbance.

Table 2 Performance indexes

Reference speed (rpm)

Control algorithm

Overshoot (%)

Settling time (ms)

Steady speed ripple (rpm)

Standard deviation of steady speed (rpm)

500 MRAC+ESO 6.6500 3 4.0000 1.4325

MRAC 7.4740 3 3.2500 1.1640 PI 5.0000 2 5.6250 2.2756

1000 MRAC+ESO 2.1100 4 3.7500 1.4708

MRAC 2.9700 4 3.3750 1.1531 PI 8.2500 20 4.7500 2.0406

2000 MRAC+ESO 0.1125 7 5.1250 1.9418

MRAC 0.6000 7 4.5000 1.6018 PI 16.6125 28 4.2500 1.6781

Fig. 19. Experimental response curves when the reference speed is 2000 rpm: (a) speed and (b) qi

∗

Fig. 20. Experimental response curves when J is increased to 11J and reference speed is 500 rpm: (a) speed and (b) qi

∗

Fig. 18. Experimental response curves when the reference speed is 1000 rpm: (a) speed and (b) qi

∗

Fig. 22. Experimental response curves when J is increased to 11J and reference speed is 2000 rpm: (a) speed and (b) qi

∗

Fig. 21. Experimental response curves when J is increased to 11J and reference speed is 1000 rpm: (a) speed and (b) qi

∗

Fig. 23. Experimental response curves in the presence of sudden load torque disturbance in the case of 500 rpm: (a) speed (MRAC). (b) qi

∗ (MRAC). (c) speed (MRAC+ESO). (d) qi∗ (MRAC+ESO).

Fig. 24. Experimental response curves in the presence of sudden load torque disturbance in the case of 1000 rpm: (a) speed (MRAC). (b) qi

∗ (MRAC). (c) speed (MRAC+ESO). (d) qi∗ (MRAC+ESO).

REFERENCES [1] G. J. Wang, C. T. Fong and K. J. Chang,

“Neural-network-based self-tuning PI controller for precise motion control of PMAC motors,” IEEE Transactions on Industrial Electronics, Vol. 48, No. 2, pp. 408-415, Apr. 2001.

[2] R. L. A. Ribeiro, A. D. Araujo, A. C. Oliveria, and C. B. Jacobina, “A high performance permanent magnet synchronous motor drive by using a robust adaptive control strategy,” Power Electronics Specialists Conference, pp. 2260-2266, 2007.

[3] S. H. Li and Z. G. Liu, “Adaptive speed control for permanent-magnet synchronous motor system with variations of load inertia,” IEEE Transactions on Industrial Electronics, Vol. 56, No. 8, pp. 3050-3059, Aug. 2009.

[4] H. H. Choi, N. T.-T. Vu and J.-W. Jung, “Digital implementation of an adaptive speed regulator for a PMSM,” IEEE Transactions on Power Electronics, Vol. 26, No. 1, pp. 3-8, Jan. 2011.

[5] T.-H. Liu, H.-T. Pu and C.-K. Lin, “Implementation of an adaptive position control system of a permanent-magnet synchronous motor and its application,” IET Electric Power applications, Vol. 4, No. 2, pp. 121-130, Feb. 2010.

[6] K.-H. Kim, “Model reference adaptive control-based adaptive current control scheme of a PM synchronous motor with an improved servo performance,” IET Electric Power applications, Vol. 3, No. 1, pp. 8-18, Jan. 2009.

[7] W.-H. Chen, “Disturbance observer based control for nonlinear system,” IEEE/ASME Transactions on

Mechatronics, Vol. 9, No. 4, pp. 706-710, Dec. 2004.

[8] S. E. Talole, J. P. Kolhe and S. B. Padke, “Extended-state-observer-based control of flexible-joint system with experimental validation,” IEEE Transactions on Industrial Electronics, Vol. 57, No. 4, pp. 1411-1419, Apr. 2010.

[9] C. K. Lai and K. K. Shyu, “A novel motor drive design for incremental motion system via sliding-mode control method,” IEEE Transactions on Industrial Electronics, Vol. 52, No. 2, pp. 499-507, Apr. 2005.

[10] S. H. Li, M. M. Zhou and X. H. Yu, “Design and implementation of terminal slide mode control method for PMSM speed regulation system,” IEEE Transactions on Industrial Informatics, Vol. 3, No. 99, pp. 1-11, Nov. 2012.

[11] K. Jezernik, J. Korelic and R. Horvat, “PMSM slide mode FPGA-based control for torque ripple reduction,” IEEE Transactions on Power Electronics, Vol. 28, No. 7, pp. 3549-3556, Jul. 2013.

[12] F. M. S. Fayez, “Robust adaptive wavelet-neural-network sliding-mode speed control for a DSP-based PMSM drive system,” Journal of Power Electronics, Vol. 10, No. 5, pp. 505-517, Jul. 2010.

[13] P. H. Cortes, M. P. Kazmierkowski, R. M. Kennel, D. E. Quevedo and J. Rodriguez, “Predictive control in power electronics and drive,” IEEE Transactions on Industrial Electronics, Vol. 55, No. 12, pp. 4312-4324, Dec. 2008.

[14] R. Errouissi, M. Ouhrouche, W.-H. Chen and A. M. Trzynadlowski, “Robust nonlinear predictive controller for a PMSM with optimized cost function,” IEEE Transactions on Industrial Electronics, Vol. 59, No. 7, pp. 2849-2858, Jul. 2012.

[15] R. J. Wai, “Hybrid fuzzy neural network control for nonlinear motor-toggle servomechanism,” IEEE Transactions on Control System Technology, Vol. 10, No. 4,

pp. 519-532, Jul. 2002.

Fig. 25. Experimental response curves in the presence of sudden load torque disturbance in the case of 2000 rpm: (a) speed (MRAC). (b) qi

∗ (MRAC). (c) speed (MRAC+ESO). (d) qi∗ (MRAC+ESO).

[16] F. J. Lin, T. S. Lee, and C. H. Lin, “Robust H∞ controller design with recurrent neural network for linear synchronous motor drive,” IEEE Transactions on Industrial Electronics, Vol. 50, No. 3, pp. 456-470, Jun. 2003.

[17] S. H. Li and H. X. Liu, “A speed control for a PMSM using finite-time feedback and disturbance compensation,” Transactions of the Institute of Measurement and Control, Vol. 32, No. 2, pp. 170-187, Apr. 2010.

[18] A. V. Sant and K. R. Rajagopal, “PM synchronous motor speed control using hybrid fuzzy-PI,” IEEE Transactions on Magnetics, Vol. 45, No. 10, pp. 4672-4675, Oct. 2009.

[19] J.-W. Jung, H. H. Choi and T.-H. Kim, “Fuzzy PD speed controller for permanent magnet synchronous motors,” Journal of Power Electronics, Vol. 11, No. 6, pp. 819-823, May. 2011.

[20] J.-W. Jung, H. H. Choi and D.-M. Lee, “Implementation of a robust fuzzy adaptive speed tracking control system for permanent magnet synchronous motors,” Journal of Power Electronics, Vol. 12, No. 6, pp. 904-911, Nov. 2012.

[21] I. D. Landau, “A survey of model reference adaptive techniques-theory and applications,” Automatica, Vol. 10, No. 4, pp. 353-379, Jun. 1974.

[22] F. Alonge and F. M. Raimondi, “Model reference adaptive control of motion control system with induction motor,” IEEE International Symposium on Industrial Electronics, pp. 803-808, 1995.

[23] N. Golea, A. Golea and M. Kadjoudj, “Robust MRAC adaptive control of PMSM drive under general parameters uncertainties,” IEEE International Conference on Industrial Technology, pp. 767-771, 2006.

[24] Y. Sozer, H. Kaufman and D. A.Torrey, “Direct model reference adaptive control of permanent magnet brushless DC motor,” IEEE International Conference on Control Applications, pp. 633-638, 1997.

[25] A. S. Kumar, M. S. Rao and Y. S. K. Babu, “Model reference linear adaptive control of DC motor using fuzzy controller,” IEEE Region 10 Conference, pp. 1603-1607, 2008

[26] N. Leena and R. Shanmugasundaram, “Adaptive controller for improved performance of brushless DC motor,” IEEE International Conference on Data Science and Engineering, pp. 117-122, 2012.

[27] A. F. A. Furtunato, “A robust control for induction motor using a variable structure model reference adaptive control (VS-MRAC),” IEEE International Power electronics Congress, pp. 61-69, 1998.

[28] H. X. Liu and S. H. Li, “Speed control for PMSM servo system using predictive functional control and extended state observer,” IEEE Transactions On Industrial Electronics, Vol. 59, No. 2, pp. 1171-1183, Feb. 2012.

[29] S. H. Li, C. J. Xia and X. Zhou, “Disturbance rejection control method for permanent magnet synchronous motor speed-regulation system,” Mechatronics, Vol. 22, No. 6, pp. 706-714, Sep. 2012.

[30] P. C. Krause, Analysis of electric machinery, New Jersey: McGraw-Hill, Chap. 4, 1995.

[31] J. Q. Han, “The extended states observer of a class of uncertain systems,” (in Chinese) Control Decision, Vol. 10, No. 1, pp. 85-88, Jan. 1995.

[32] J. Q. Han, “From PID to active disturbance rejection control,” IEEE Transactions on Industrial Electronics, Vol. 56, No. 3, pp. 900-906, Mar. 2009.

[33] R. Miklosovic and Z. Q. Gao, “A robust two-degree-of-fredom control design technique and its

practical application,” IEEE Industry Applications Conference, pp. 1495-1502, 2004.

[34] H. K. Khalil. Nonlinear Systems, Prentice-Hall, Chap. 4, 2002.

[35] H. B. Sun, S. H. Li and S. M. Fei, “A composite control scheme for 6DOF spacecraft formation control,” Acta Astronautica, Vol. 69, No. 7-8, pp. 595-611, Sep. 2011.

[36] J. Su, W. Qiu, H. Ma and P.-Y. Woo, “Calibration-free robotic eye-hand coordination based on an auto disturbance-rejection controller,” IEEE Transactions on Robotics, Vol. 20, No. 5, pp. 899-907, Oct. 2004.

[37] D. Sun, “Comments on active disturbance rejection control,” IEEE Transactions on Industrial Electronics, Vol. 54, No. 6, pp. 3428-3429, Dec. 2007.

Xiaodi Li was born in Wuhu, China, in 1989. He received the B.S. degree in Automatic Control from Hohai University, Nanjing, China, in 2011 and the M.S. degree in Automatic Control from Southeast University, Nanjing, in 2014. His current research interests include control theory and its applications in motor

system.

Shihua Li was born in Pingxiang, China, in 1975. He received the B.S., master, Ph.D. degrees all in Automatic Control from Southeast University, Nanjing, China in 1995, 1998 and 2001, respectively. Since 2001, he has been with School of Automation, Southeast University, where he is currently a professor.

His main research interests include nonlinear control theory with applications to robot, spacecraft, AC motor and other mechanical systems.